Algebra and Logic

, Volume 50, Issue 1, pp 46–61

Growth in Poisson algebras


DOI: 10.1007/s10469-011-9123-z

Cite this article as:
Ratseev, S.M. Algebra Logic (2011) 50: 46. doi:10.1007/s10469-011-9123-z

A criterion for polynomial growth of varieties of Poisson algebras is stated in terms of Young diagrams for fields of characteristic zero. We construct a variety of Poisson algebras with almost polynomial growth. It is proved that for the case of a ground field of arbitrary characteristic other than two, there are no varieties of Poisson algebras whose growth would be intermediate between polynomial and exponential. Let V be a variety of Poisson algebras over an arbitrary field whose ideal of identities contains identities {{x1, y1}, {x2, y2}, . . . , {xm, ym}} = 0 and {x1, y1} · {x2, y2} · . . . · {xm, ym} = 0, for some m. It is shown that the exponent of V exists and is an integer. For the case of a ground field of characteristic zero, we give growth estimates for multilinear spaces of a special form in varieties of Poisson algebras. Also equivalent conditions are specified for such spaces to have polynomial growth.


Poisson algebragrowth of varietycolength of variety

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  1. 1.Ul’yanovsk State UniversityUl’yanovskRussia